# Comparison theorem (algebraic geometry)

A theorem on the relations between homotopy invariants of schemes of finite type over the field $\mathbf C$ in classical and étale topologies.

Let $X$ be a scheme of finite type over $\mathbf C$, while $F$ is a constructible torsion sheaf of Abelian groups on $X _ {\textrm{ et } }$. Then $F$ induces a sheaf on $X$ in the classical topology, and there exist canonical isomorphisms

$$H ^ {q} ( X _ {\textrm{ et } } , F) \cong \ H ^ {q} ( X _ {\textrm{ class } } , F).$$

On the other hand, a finite topological covering of a smooth scheme $X$ of finite type over $\mathbf C$ has a unique algebraic structure (Riemann's existence theorem). The fundamental étale group of $X _ {\textrm{ et } }$[1] is therefore the pro-finite completion of the ordinary group of classes of homotopically equivalent loops:

$$\pi _ {1} ( X _ {\textrm{ et } } ) = \ [ \pi _ {1} ( X _ {\textrm{ class } } )] \widehat{ {}} .$$

Moreover, if $X _ {\textrm{ class } }$ is simply connected, then $X _ {\textrm{ et } } = \widehat{X} _ { \mathop{\rm cl} }$, where $X _ { \mathop{\rm cl} }$ and $X _ {\textrm{ et } }$ are the classical and étale homotopy types of the scheme $X$, respectively (see [1], [2]).

#### References

 [1] M. Artin, "The étale topology of schemes" , Proc. Internat. Congress Mathematicians (Moscow, 1966) , Mir (1968) pp. 44–56 [2] D. Sullivan, "Geometric topology" , M.I.T. (1971) (Notes)
How to Cite This Entry:
Comparison theorem (algebraic geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Comparison_theorem_(algebraic_geometry)&oldid=46413
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article